Optimization methods m 5770-001 /6640-001

HW 6

November 14, 2016

1. Write a code for Nelder-Mead method. Use http://www.scholarpedia.org/article/Nelder-Mead algorithm. Test the code by minimizing

f(x) = x_{1}^{2} + x_{2}^{2} + x_{1}x_{2} + x_{1}x_{3} + x_{3}^{2}

2. Test Nelder-Mead method, a quasi-Newton method, and a conjugate gradient method minimizing the following functions: (see https://en.wikipedia.org/wiki/Test functions for optimization)

2a. Beale’s function

b(x, y) = (1.5 − x + xy)^{2} + (2.25 − x + xy^{2})^{2} + (2.625 − x + xy^{3})^{2}

2b. Rosenbrock function, n=4

f(x) = ∑_{i=1}^{n−1} [100(x_{i+1} − x_{i}^{2})^{2} + (x_{i} − 1)^{2}]

Compare the results obained by these methods.

3. Using Lagrange multipliers method, find

min_{x∈R^{n}} (∑_{i=1}^{n} a_{i}^{2} x_{i}) subject to a constraint: ∑_{i=1}^{n} \frac{1}{x_{i} − c} = \frac{1}{b}